### An uniform boundedness for Bochner-Riesz operators related to the Hankel transform.

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One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by $${\left(\frac{2}{\lambda {e}^{t}+1}\right)}^{\alpha}{e}^{xt}=\sum _{n=0}^{\infty}{\mathcal{E}}_{n}^{\left(\alpha \right)}(x;\lambda )\frac{{t}^{n}}{n!}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}\lambda \in \u2102\setminus \{-1\}\phantom{\rule{0.166667em}{0ex}},$$ and as an “exceptional family”...

Let w be a generalized Jacobi weight on the interval [-1,1] and, for each function f, let Sf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators S in some weighted L spaces. The study of the norms of the kernels K related to the operators S allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.

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